Virtual Knot Cobordism and the Affine Index Polynomial

TL;DR

This paper introduces the affine index polynomial as a new invariant for virtual knots, demonstrating its invariance under cobordism and concordance, and applying it to determine the four-ball genus of virtual knots.

Contribution

It defines and proves the affine index polynomial as a concordance invariant for virtual knots and links, expanding tools for virtual knot classification.

Findings

Affine index polynomial is a concordance invariant.

The polynomial is invariant under certain cobordisms.

Applications to four-ball genus of virtual knots.

Abstract

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.